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We designed a superconducting front-end circuit for an analog-to-digital converter and experimentally evaluated its signal-to-noise-ratio (SNR) at a sampling frequency of 10 GHz. The modulator was based on a first-order sigma-delta modulation with an LR integrator. Correct noise shaping was experimentally confirmed, and an SNR of 70.9 dB at a bandwidth of 10 MHz was achieved. In the circuit design, we investigated the effect of leakage in the LR integrator and thermal noise on the SNR by using transfer function analysis and circuit simulations with noise sources. Circuit parameters were designed to keep the discrepancy of the SNR from the ideal value (77.6 dB) within 6 dB. In the experimental evaluation, the modulated data signal from the superconducting chip was accumulated with room-temperature electronics. A fast Fourier transform (FFT) calculation was performed to obtain power spectra and SNRs. The power spectra and SNRs were consistent with the predictions of the transfer function analysis and circuit simulations.